Generalized Gibbs ensemble in integrable lattice models

By L. Vidmar, Marcos Rigol. 2016

602 citations recorded by the source corpus.

The generalized Gibbs ensemble (GGE) was introduced ten years ago to describe observables in isolated integrable quantum systems after equilibration. Since then, the GGE has been demonstrated to be a powerful tool to predict the outcome of the relaxation dynamics of few-body observables in a variety of integrable models, a process we call generalized thermalization. This review discusses several fundamental aspects of the GGE and generalized thermalization in integrable systems. In particular, we focus on questions such as: which observables equilibrate to the GGE predictions and who should play the role of the bath; what conserved quantities can be used to construct the GGE; what are the differences between generalized thermalization in noninteracting systems and in interacting systems mappable to noninteracting ones; why is it that the GGE works when traditional ensembles of statistical mechanics fail. Despite a lot of interest in these questions in recent years, no definite answers have been given. We review results for the XX model and for the transverse field Ising model. For the latter model, we also report original results and show that the GGE describes spin–spin correlations over the entire system. This makes apparent that there is no need to trace out a part of the system in real space for equilibration to occur and for the GGE to apply. In the past, a spectral decomposition of the weights of various statistical ensembles revealed that generalized eigenstate thermalization occurs in the XX model (hard-core bosons). Namely, eigenstates of the Hamiltonian with similar distributions of conserved quantities have similar expectation values of few-spin observables. Here we show that generalized eigenstate thermalization also occurs in the transverse field Ising model.

Subfields: Atomic and Molecular Physics, and Optics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/da8b36f9f0e821119bc1115aa6ddea78c412ea69

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Citations in the corpus

Aspects of non-equilibrium in classical and quantum systems: Slow relaxation and glasses, dynamical large deviations, quantum non-ergodicity, and open quantum dynamics (2017, background): “An exception to the above scenario are integrable systems [64] which equilibrate to a generalised Gibbs ensemble [72, 73].”
Entanglement dynamics after quantum quenches in generic integrable systems (2017, background): “that the reduced density matrix for long times is not thermal. Remarkably, a statistical description of local properties of the steady state is possible in terms of a Generalised Gibbs Ensemble (GGE) [12,16,17,39{66], which is obtained by complementing the Gibbs ensemble with all the local and quasi”
Prethermalization and Thermalization in Isolated Quantum Systems (2018, background): “However, because of eigenstate thermalization in quantum chaotic systems, and its generalized version in integrable systems, one can equally well use traditional ensembles of statistical mechanics for the projected dynamics when the reference dynamics is quantum chaotic [22] and generalized Gibbs en”
Complex Spacing Ratios: A Signature of Dissipative Quantum Chaos (2019, background): “After a quench, local observables of chaotic systems thermalize to values that can be predicted by a thermodynamic ensemble average [100].”
Nonequilibrium quantum dynamics and transport: from integrability to many-body localization (2016, background): “to a generalized Gibbs ensemble [10] that includes more conservation laws than just energy).”
Statistical localization: From strong fragmentation to strong edge modes (2019, background): “[78] Lev Vidmar and Marcos Rigol, “Generalized gibbs ensemble in integrable lattice models,” Journal of Statistical Mechanics: Theory and Experiment 2016, 064007”
Entanglement Spreading in a Minimal Model of Maximal Many-Body Quantum Chaos (2018, neutral): “On the other hand, for h 1⁄4 0 the system is integrable, and finite subsystems relax to generalized Gibbs ensembles [71,72].”
Emergent Quantum State Designs from Individual Many-Body Wave Functions (2021, background): “and hence will not form 1-designs at effective infinite temperature [56].”
Ergodicity breaking transition in finite disordered spin chains (2020, background): “A well-established nonergodic behavior in translationally invariant systems occurs at integrable points [8– 16], for which eigenstates do not obey ETH [17–25] and the long-time averages of observables after quantum quenches are described by the generalized Gibbs ensemble [23, 26].”
Localization from Hilbert space shattering: From theory to physical realizations (2020, background): “[17] Lev Vidmar and Marcos Rigol, “Generalized gibbs ensemble in integrable lattice models,” Journal of Statistical Mechanics: Theory and Experiment 2016, 064007 (2016).”
Drude Weight for the Lieb-Liniger Bose Gas (2017, background): “(3.4) A generalized Gibbs ensemble (GGE) is specified by some density function f(v) independent of x. Microscopically this means that the hard rods have uniform density and independent velocities with probability density function ρ−1f(v).”